# Proving postulate about a property fo spherical vectors

Assume we have $$X, Y$$ constant unit vectors of $$\mathbb{R}^3$$

I postulate that the maximum of the function:

$$(V \cdot X) (V \cdot Y)$$

I reached by the halfway vector between $$X,Y$$ i.e at the vector $$V_0 = slerp(X,Y, 0.5)$$

To try to prove it I tried finding the critical point of the derivative, i.e:

$$(V'\cdot X)(V\cdot Y) + (V\cdot X)(V'\cdot Y)$$

But that is leading me down a rabbit hole I don't seem to be able to get out of.

• Derivative with respect to $V$? But you'd better constrain $V$ to be a unit vector as well. – Ted Shifrin Jun 5 '20 at 3:50

Without loss of generality, choose $$X=(1,0,0)$$ and $$Y=(\cos\phi_0,\sin\phi_0,0)$$. Then (assuming $$V$$ is also unit vector), you can write $$V$$ in polar coordinates as $$V=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$$. Then your expression becomes $$\sin\theta\cos\phi(\sin\theta\cos\phi\cos\phi_0+\sin\theta\sin\phi\sin\phi_0)=\sin^2\theta\cos\phi\cos(\phi-\phi_0)$$ If you want the maximum, you get $$\theta=\pi/2$$, so it's in the same plane. Also $$\frac d{d\phi}\cos\phi\cos(\phi-\phi_0)=-\sin(2\phi-\phi_0)=0$$ so $$\phi=\frac{\phi_0}2$$ You will need to consider separately the case where $$\phi_0=\pi$$